Question

19 find the value for ad.

Explanation:

Step1: Identify the theorem

This is a case of the Basic Proportionality Theorem (Thales' theorem), which states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So, we have \(\frac{AB}{AD}=\frac{BC}{DE}\).

Step2: Define the segments

Let \(AB = 15\), \(AD=x + 12\), \(BC = 4\), and \(DE=10\). Substituting into the proportion: \(\frac{15}{x + 12}=\frac{4}{10}\)

Step3: Cross - multiply

Cross - multiplying gives us \(4(x + 12)=15\times10\)

Step4: Expand and solve for x

First, expand the left - hand side: \(4x+48 = 150\)
Subtract 48 from both sides: \(4x=150 - 48=102\)
Divide both sides by 4: \(x=\frac{102}{4}=25.5\)

Step5: Find AD

Now, \(AD=x + 12\). Substitute \(x = 25.5\) into the expression for \(AD\): \(AD=25.5+12 = 37.5\)

Answer:

\(37.5\)